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Derivative

This page is meant to serve as a quick overview of the basics of derivitives.

Derivitives

The derivative of a function is f'(x) = lim as h approaches 0 (f(x+h)-f(x/h)), provided that the limit exist.

Alternative Approach

f'(a) = lim x ⇒ a ( f(x) - F(a) ) / (x-a)

Differentiability

For any acute angle Θ in standard position, the following applies:

 A corner A cusp A verticle tangent A discontinuity If f has a derivative at x=a, then f is continuous at x=a.

Derivative Rules

 Derivative of a Constant If c is a real number, the derivative of a constant function is 0. d/dx[c] = 0 Sum and Difference Rules The sum or difference of any two differentiable functions is differentiable and is the sum or difference of their derivatives. d/dx[f(x) + g(x)] = f'(x) + g'(x) d/dx[f(x) - g(x)] = f'(x) - g'(x) Constant Multiple Rule If f is a differentiable function and c is a constant, then cf is also differentiable d/dx[cf(x)] = cf'(x) Power Rule If n is a rational number, then the function f(x) = xn is differentiable. d/dx[xn] = nxn-1 Product Rule The product of two differentiable functions, f and g, is differentiable. d/dx[f(x)g(x)] = f(x)g'(x) + g(x)f'(x) Quotient Rule The quotient f/g, of two differentiable functions, f and g, is differentiable at all values of x for which g(x) ≠ 0. d/dx[ f(x)/g(x) ] = (g(x)f'(x) - f(x)g'(x)) / [g(x)]2 Chain Rule No Explanation Currently d/dx[f(g(x))] = f'(g(x))g'(x) Power Rule If c is a real number, the derivative of a constant function is 0. d/dx[f(g(x))] = f'(g(x))g'(x)

Derivatives of Trig Functions

 d/dx[sinx] = cosx d/dx[cosx] = -sinx d/dx[tanx] = sec2x d/dx[cscx] = -csc x cot x d/dx[secx] = sec x tan x d/dx[cotx] = -csc2x

Jerk

Jerk is the derivative of acceleration. Jt = da/dt = d3s / dt3

Implicit Differentiation

Used to find dy/dx when it is hard to find what y equals.

Inverse Trigonometric

 d/dx arcsin x  = 1  √(1 - x2)

 d/dx arccsc x = -1  |x| √(x2 - 1)

 d/dx arccos x = -1  √(1 - x2)

 d/dx arcsec x = 1  |x| √(x2 - 1)

 d/dx arctan x = 1  1 + x2

 arccot x = -1  1 + x2